Dokapon: Monster Hunter (2001)

Dokapon is a deep, long playing, thoroughly absorbing RPG that takes place on Dokkano Island in the town of Poponga, a mystical place full of mystical creatures. Your goal is to become a a licensed adventurer - to do this you must pass an adventurer's exam by completing a number of assigned missions, and along the way defeat all kinds of monsters in battle. Once beaten, you can collect the monster and make it your "monster partner", and use it in future battles to defeat opponents.